Irrational to be entertained by Dynasty recording (6)
I believe the answer is:
taping
'recording' is the definition.
(I know that taping is a type of recording)
'irrational to be entertained by dynasty' is the wordplay.
'irrational' becomes 'pi' (an irrational number).
'to be entertained by' indicates putting letters inside (inserted letters are hosted or entertained).
'dynasty' becomes 'tang' (Chinese dynasty).
'pi' going inside 'tang' is 'TAPING'.
(Other definitions for taping that I've seen before include "Fastening" , "Wrapping" , "Recording" , "Securing" .)